Let be a complex reductive group and a -module. Then the th jet scheme acts on the th jet scheme for all . We are interested in the invariant ring and whether the map induced by the categorical quotient map is an isomorphism, surjective, or neither. Using Luna’s slice theorem, we give criteria for to be an isomorphism for all , and we prove this when , , , or and is a sum of copies of the standard module and its dual, such that is smooth or a complete intersection. We classify all representations of for which is surjective or an isomorphism. Finally, we give examples where is surjective for but not for finite , and where it is surjective but not injective.
Soient un groupe réductif complexe et un -module. Alors , le schéma des jets d’ordre de , opère dans , le schéma des jets d’ordre de , pour tout . Nous nous intéressons à l’anneau des invariants et au morphisme induit par le morphisme du quotient catégorique : ce morphisme est-il un isomorphisme, surjectif, ou non ? En utilisant le théorème du slice de Luna, nous obtenons des critères pour que soit un isomorphisme pour tout . Nous montrons que c’est bien le cas lorsque , , , ou et est un somme directe de copies du module standard et de son dual, pourvu que soit lisse ou une intersection complète. Nous classifions toutes les représentations de telles que soit surjectif ou un isomorphisme. Enfin, nous donnons des exemples où est surjectif pour mais non surjectif pour fini, et d’autres exemples où est surjectif mais non injectif.
Accepted:
Published online:
DOI: 10.5802/aif.2996
Classification: 13A50, 14L24, 14L30
Keywords: jet schemes, classical invariant theory
@article{AIF_2015__65_6_2571_0, author = {Linshaw, Andrew R. and Schwarz, Gerald W. and Song, Bailin}, title = {Jet schemes and invariant theory}, journal = {Annales de l'Institut Fourier}, pages = {2571--2599}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {65}, number = {6}, year = {2015}, doi = {10.5802/aif.2996}, zbl = {1342.13009}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2996/} }
TY - JOUR TI - Jet schemes and invariant theory JO - Annales de l'Institut Fourier PY - 2015 DA - 2015/// SP - 2571 EP - 2599 VL - 65 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2996/ UR - https://zbmath.org/?q=an%3A1342.13009 UR - https://doi.org/10.5802/aif.2996 DO - 10.5802/aif.2996 LA - en ID - AIF_2015__65_6_2571_0 ER -
Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin. Jet schemes and invariant theory. Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2571-2599. doi : 10.5802/aif.2996. https://aif.centre-mersenne.org/articles/10.5802/aif.2996/
[1] Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1998, pp. 1-32 | MR: 1672108 | Zbl: 0963.14015
[2] Chiral algebras, American Mathematical Society Colloquium Publications, Tome 51, American Mathematical Society, Providence, RI, 2004, vi+375 pages | MR: 2058353 | Zbl: 1138.17300
[3] Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A., Tome 83 (1986) no. 10, pp. 3068-3071 | Article | MR: 843307 | Zbl: 0613.17012
[4] Singularités rationnelles et quotients par les groupes réductifs, Invent. Math., Tome 88 (1987) no. 1, pp. 65-68 | Article | MR: 877006 | Zbl: 0619.14029
[5] An introduction to motivic integration, Strings and geometry (Clay Math. Proc.) Tome 3, Amer. Math. Soc., Providence, RI, 2004, pp. 203-225 | MR: 2103724 | Zbl: 1156.14307
[6] Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., Tome 135 (1999) no. 1, pp. 201-232 | Article | MR: 1664700 | Zbl: 0928.14004
[7] Geometry on arc spaces of algebraic varieties, European Congress of Mathematics, Vol. I (Barcelona, 2000) (Progr. Math.) Tome 201, Birkhäuser, Basel, 2001, pp. 327-348 | MR: 1905328 | Zbl: 1079.14003
[8] Invariants of -jet actions, Houston J. Math., Tome 10 (1984) no. 2, pp. 159-168 | MR: 744898 | Zbl: 0568.14007
[9] Jet schemes and singularities, Algebraic geometry—Seattle 2005. Part 2 (Proc. Sympos. Pure Math.) Tome 80, Amer. Math. Soc., Providence, RI, 2009, pp. 505-546 | Article | MR: 2483946 | Zbl: 1181.14019
[10] Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, Tome 88, American Mathematical Society, Providence, RI, 2001, xii+348 pages | Article | MR: 1849359 | Zbl: 0981.17022
[11] Vertex operator algebras and the Monster, Pure and Applied Mathematics, Tome 134, Academic Press, Inc., Boston, MA, 1988, liv+508 pages | MR: 996026 | Zbl: 0674.17001
[12] The Nash problem on arc families of singularities, Duke Math. J., Tome 120 (2003) no. 3, pp. 601-620 | Article | MR: 2030097 | Zbl: 1052.14011
[13] Vertex algebras for beginners, University Lecture Series, Tome 10, American Mathematical Society, Providence, RI, 1998, vi+201 pages | MR: 1651389 | Zbl: 0924.17023
[14] Differential algebra and algebraic groups, Academic Press, New York-London, 1973, xviii+446 pages (Pure and Applied Mathematics, Vol. 54) | MR: 568864 | Zbl: 0264.12102
[15] String cohomology, 1995 (Lecture at Orsay)
[16] Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984, x+308 pages | Article | MR: 768181 | Zbl: 0569.14003
[17] Arc spaces and the vertex algebra commutant problem, Adv. Math., Tome 277 (2015), pp. 338-364 | Article | MR: 3336089
[18] Motivic measures, Astérisque (2002) no. 276, pp. 267-297 (Séminaire Bourbaki, Vol. 1999/2000) | Numdam | MR: 1886763 | Zbl: 0996.14011
[19] Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, p. 81-105. Bull. Soc. Math. France, Paris, Mémoire 33 | Numdam | MR: 342523 | Zbl: 0286.14014
[20] Chiral de Rham complex. II, Differential topology, infinite-dimensional Lie algebras, and applications (Amer. Math. Soc. Transl. Ser. 2) Tome 194, Amer. Math. Soc., Providence, RI, 1999, pp. 149-188 | MR: 1729362 | Zbl: 0999.17037
[21] Chiral de Rham complex, Comm. Math. Phys., Tome 204 (1999) no. 2, pp. 439-473 | Article | MR: 1704283 | Zbl: 0952.14013
[22] Jet schemes of locally complete intersection canonical singularities, Invent. Math., Tome 145 (2001) no. 3, pp. 397-424 (With an appendix by David Eisenbud and Edward Frenkel) | Article | MR: 1856396 | Zbl: 1091.14004
[23] Arc structure of singularities, Duke Math. J., Tome 81 (1995) no. 1, p. 31-38 (1996) (A celebration of John F. Nash, Jr.) | Article | MR: 1381967 | Zbl: 0880.14010
[24] Representations of simple Lie groups with regular rings of invariants, Invent. Math., Tome 49 (1978) no. 2, pp. 167-191 | Article | MR: 511189 | Zbl: 0391.20032
[25] The global sections of the chiral de Rham complex on a Kummer surface (http://arxiv.org/abs/1312.7386)
[26] Arc spaces, motivic integration and stringy invariants, Singularity theory and its applications (Adv. Stud. Pure Math.) Tome 43, Math. Soc. Japan, Tokyo, 2006, pp. 529-572 | MR: 2325153 | Zbl: 1127.14004
[27] The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939, xii+302 pages | MR: 1488158 | Zbl: 1024.20502
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